Locally Quasi-Convex Compatible Topologies on a Topological Group

Abstract

For a locally quasi-convex topological abelian group (G, τ), we study the poset C (G, τ) of all locally quasi-convex topologies on G that are compatible with τ (i.e., have the same dual as (G, τ)) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G, Gb). Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates “from below”, our strategy consists of finding appropriate subgroups H of G that are easier to handle and show that C (H) and C (G/H) are large and embed, as a poset, in C (G, τ). Important special results are: (i) if K is a compact subgroup of a locally quasi-convex group G, then C (G) and C (G/K) are quasi-isomorphic (3.15); (ii) if D is a discrete abelian group of infinite rank, then C (D) is quasi-isomorphic to the poset FD of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group G with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset C (G) is as big as the underlying topological structure of (G, τ) (and set theory) allows. For a metrizable connected compact group X, the group of null sequences G = c0(X) with the topology of uniform convergence is studied. We prove that C (G) is quasi-isomorphic to P(R) (6.9).